This is actually a question that many Judges and legal people have been debating.
First of all, the VP and Pres cannot be from the same state. So Bill Clinton would have to relocate prior to being nominated for the position.
As far as him becoming VP, there is no clear cut answer -- and the Supreme Court has not ruled on the matter -- since it has never been relevant.
The Constitution says that to be President, you must be 35 years old, born in the US, be a US citizen, and have lived in the US for at least the past 14 years. The 12th amendment says that those rules apply to the VP as well.
Fine, Clinton meets those standards.
Next, there is the 22nd amendment, which says that a President can't be elected more than twice. This is the "sticky" part, since Bill Clinton has already been elected twice, and the 12th amendment says that the President rules also apply to the VP.
Many, but not all, experts
believe that it would be okay for him to run as the VP candidate. If anything happened to Hillary, he could take over as President, but he could not run again in a future election. However, others believe that it would not be allowed, since it would violate a combination of the 12th and 22nd amendments.
If she decided to nominate him, I am quite sure that the opponent would challenge it, likely to the Supreme Court.
Personally, I believe that she could nominate him to be VP, but I don't think she would. However, if she becomes President, I think it is more likely that she would try to appoint him to be a Supreme Court Justice (a former President, ***** ***** Taft, was on the Supreme Court after serving as President).
Let me know if you have any questions. If not, thanks for pressing "Accept".